What is the smallest positive number that is prime and $10$ less than a perfect square?
Solution: A number 10 less than an even perfect square cannot be prime, so let's check odd perfect squares greater than 10:

$\bullet$ $5^2=25,$ $25-10=15,$ composite.

$\bullet$ $7^2=49,$ $49-10=39,$ composite.

$\bullet$ $9^2=81,$ $81-10=71.$

Checking prime numbers up to $7$ $($the  largest prime less than $\sqrt{71})$ as potential divisors, we  see that $71$ is prime. Thus, the smallest prime that is $10$ less  than a perfect square is $\boxed{71}.$